| L(s) = 1 | + 8·7-s − 8·19-s + 4·25-s + 8·31-s + 8·37-s + 12·49-s + 8·103-s − 56·109-s + 28·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1.83·19-s + 4/5·25-s + 1.43·31-s + 1.31·37-s + 12/7·49-s + 0.788·103-s − 5.36·109-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s + 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.587239819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587239819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 2 T^{2} + 33 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 14 T^{2} + 129 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T^{2} + 762 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 14 T^{2} - 351 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 8 T^{2} + 1050 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 50 T^{2} + 3825 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 64 T^{2} + 3570 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 8 T^{2} - 1398 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T^{2} + 3030 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 128 T^{2} + 10410 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 136 T^{2} + 11010 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 266 T^{2} + 27753 T^{4} - 266 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 88 T^{2} + 2226 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 208 T^{2} + 22146 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 152 T^{2} + 13722 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 146 T^{2} + 13233 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 172 T^{2} + 21606 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.59995239759796384575443723258, −3.40115118862095487011183889496, −3.37937468640842531682727621720, −3.22154397471871276366180249914, −3.10738408309340640306190959928, −3.02104588870076831572568379640, −2.91175835356353368399335585846, −2.83243655251238924370354528644, −2.53056085332672203852734687000, −2.52390058660053924227946943497, −2.46271910114166375219419299602, −2.14174526118513057126558490459, −2.12615392125778918629097018997, −1.89998751636591855723660198773, −1.87774330177259917140852766999, −1.69354537167868441863769487856, −1.68130484232095230832576882433, −1.60579689076358203675037801330, −1.19749245138678882116307096611, −1.16429850444188793512273569487, −1.03064429535488964589658588265, −0.805454613890060925437154469751, −0.61349949798061218896706797146, −0.50357090938047637364049492536, −0.06780184692659005415246426677,
0.06780184692659005415246426677, 0.50357090938047637364049492536, 0.61349949798061218896706797146, 0.805454613890060925437154469751, 1.03064429535488964589658588265, 1.16429850444188793512273569487, 1.19749245138678882116307096611, 1.60579689076358203675037801330, 1.68130484232095230832576882433, 1.69354537167868441863769487856, 1.87774330177259917140852766999, 1.89998751636591855723660198773, 2.12615392125778918629097018997, 2.14174526118513057126558490459, 2.46271910114166375219419299602, 2.52390058660053924227946943497, 2.53056085332672203852734687000, 2.83243655251238924370354528644, 2.91175835356353368399335585846, 3.02104588870076831572568379640, 3.10738408309340640306190959928, 3.22154397471871276366180249914, 3.37937468640842531682727621720, 3.40115118862095487011183889496, 3.59995239759796384575443723258
Plot not available for L-functions of degree greater than 10.
